Homogeneous S-Lemma and its application to asymptotic stability of a class of switched nonlinear systems

This paper extends the strict S-Lemma proposed by Yakubovich and uses the improved strict S-Lemma to investigate the asymptotic stability of a class of switched nonlinear systems. First, the strict S-Lemma is extended from quadratic forms to homogeneous polynomials, where the improved S-Lemma is named the strict homogeneous S-Lemma (short for the SHS-Lemma). Then by utilizing the SHS-Lemma, it is proved that a switched nonlinear polynomial system with two sub-systems admits a Lyapunov function with homogeneous derivative (short for LFHD) if and only if it has a convex combination of the vector fields of its two sub-systems that admits a LFHD. Furthermore, it is shown that the “if” part of the former property still holds for switched polynomial systems with three or more sub-systems but the “only if” part does not even for switched linear systems.